Understanding the structural, electro-magnetic, mechanical, thermal and transport properties of Strontium based perovskite (SrPrO3)

With the help of density functional theory, we have explored the structural, electro-magnetic, mechanical, thermal and transport properties of SrPrO3 perovskite. The optimization of energy reveals that the alloy is stable in ferromagnetic phase. The electronic and magnetic properties were calculated by generalized gradient approximation (GGA), Hubbard approximation (GGA + U) and Tran–Blaha modified Becke–Johnson (GGA+mBJ) potentials methods. The electronic structure depicts the half-metallic nature for the material in both the GGA and GGA+U approximations. The semiconducting nature is revealed by mBJ approximation. The elastic parameters reveal the ductile nature of the material. The response towards temperature and pressure of the different thermal parameters have also been evaluated with the help of quasi-harmonic Debye model. The thermoelectric potency of the material is measured in terms of Seebeck coefficient (S), electrical conductivity (σ/τ), thermal conductivity (κ), power factor and figure of merit (ZT). On the whole, these properties pose the possible applications of this material towards solid state device and thermoelectric sphere.


Introduction
As there is an intense research going on to search some new multi-functional materials that can be utilized for different purposes.In this category perovskites and its derivatives are in spotlight because of its various properties like ferroelectricity [1], colossal magnetoresistance [2], charge-ordering, superconductivity, multiferroicity, half-metallicity etc Keeping in view they are proven suitable candidates for multifunctional devices [3] and always draw massive attention from the scientific community.As a result, to spot a new material from this class always attract researchers.Among all, perovskite oxides, especially f-electron-based, have been widely acknowledged because most of them disclose the half-metallic nature due to the interactions of localised f-electronic states at the Fermi level, resulting in an appealing electronic profile with strong spin polarisation and significant magnetism.Spin polarization is the degree to which the total spin magnetic moment of a system is oriented in a particular direction.The asymmetric density of states distribution leads to non-zero spin polarization at the Fermi level.The directional and coherent electron spin motion constitutes the spin current, which carries information in spintronics devices.Spin currents don't dissipate heat energy, as a result, resolve the heat problems that are frequently encountered by electronic devices.Hence, have applications in spintronics.Also, in today's time, surge in energy demand and environmental deterioration are the major issues to be resolved for which scientific community is looking for an alternative.As an alternative thermoelectricity is one of the options which converts waste heat into useful energy.Many rare-earth-based perovskites have undergone extensive scrutiny to understand their thermoelectric accomplishments.For electronic optical and thermoelectric properties PrMO3 (M = Al, Ga and In) have been recently investigated [4].
Zahid et al examined the physical properties of BaXO 3 (X = Pr, U) and predicted the half-metallic nature of these alloys [5].Also, Sajad et al studied SrAmO 3 and reported the alloy as thermodynamically stable and also possess half-metallic nature [6].Sr2HoNbO6 [7] Sr2ReEuO6 [8] gives good thermoelectric output.Recently, Khandy et al and Dar et al investigated a series of materials from this class specifically, BaNpO3 [9], BaPuO3 [10], and SrPuO3 [11], and they came up with the conclusion that these compounds have a decisively half-metallic nature and are potential candidate for spintronics and green energy generation.These literature findings suggest us that these materials are suitable for spintronics and thermoelectric applications.So, in terms of the cited literature, we have carried the same DFT calculations to investigated the various physical properties of simple perovskite i.e.SrPrO 3 .

Computational details
First principle DFT calculations were suitably carried out to evaluate the various physical properties of SrPrO 3 alloy.For present study, the basis set chosen is the linearized augmented plane wave with full-potential formulism (FP-LAPW) embedded in Wien2k code [12].We have used GGA [13], GGA+U [14] and mBJ [15] to treat exchange correlation term.The unit cell is divided into two parts i.e. muffin tin spheres and interstitial space.The muffin tin radii of Sr, Pr and O atoms are 2.5, 2.09 and 1.71 (a.u.) respectively.The plane-wave cut-off parameter K R MT Max = 7.0 is used while, the angular momentum vector l Max = 10 is adopted.We ensured that the charge and energy were fully converged.To run self-consistent field calculations, we have employed the tetrahedral method and a k-mesh of 3000 points in the Brillouin zones (BZ).To determine the mechanical strength of the alloy, we have used cubic-elastic package incorporated in Wien2k [16].Subsequently, the exhibition of spin polarisation which essentially descripts the asymmetric density of states distribution pinned around the fermi level (E F ) creates a magnetic moment of this alloy.The thermodynamic properties have been investigated by using quasi-harmonic Debye model [17] while semiclassical Boltzmann theory within Boltz Trap scheme [18] has been used to calculate the different transport parameters.

Structural properties
SrPrO 3 perovskite type structure crystalizes in a cubic along with Pm-3m (221) space symmetry having positional Wyckoff coordinates of corresponding atoms where Sr is located at origin having coordinates (0,0,0), Pr resides at (0.5,0.5,0.5) and O sited at (0.5,0.5,0).So, this defines the complete structure of the unit cell as displayed in figure 1. Next, we have tried to see the stability of this perovskite type structure in terms of their ground state energies for which Birch Murnaghan's equation [19] of state has been employed to figure out their total energies.The structural optimisation in two different phases particularly ferromagnetic and non-magnetic, affirms that the material stabilises in ferromagnetic phase as displayed in figure 2. The optimized values of lattice parameter (Å), minimum volume (V 0 ), bulk modulus B (GPa), its derivative (B') and minimum energy (Ry), has been enlisted in table 1.The reliability of the obtained lattice constant can be also verified from the empirical relation enumerated as below: Where, α = 0.06741, β = 0.4905 and γ = 1.2921 are constants and r A and r B be the radius of cations and r O is radius of anion [20].The estimated lattice constant obtained from the given relation is in good agreement with the optimized lattice constant.

Formation energy
In order to analyse the stability of the alloy we have calculated the formation energy (H f ) with the help of relation: , where, E total is the energy released by compound, while, E , Sr E Pr and E O are the energies released by the lattices of Sr, Pr and O, respectively.The a, b, c represents number of atoms for the respective unit cells [21].It measures the difference between the optimized energy and the energy of its constituent atoms.A positive value of H f denotes a spontaneous and unstable nature, whereas a negative value flash the thermodynamically stability of the alloy [22].Here, in case of SrPrO 3 alloy the value of formation energy is −2.84 eV atom −1 which confirms stability of the material hence, it implies that it can be synthesized experimentally.

Electronic and magnetic properties
Electronic properties are highly desired in understanding the material for its diverse applications.These properties are evaluated from the band structure and density of states (total and partial) and for this relaxed structural parameters are used.At first, the electronic band structure as shown in figure 3, has been predicted from the GGA approximation which discloses the half-metallic nature of the alloy where, spin up displays the metallic nature as energy levels are present at fermi level while semiconducting nature is depicted in spin down channel with an indirect band gap of 2.43 eV at Γ-M symmetry points.However, we have used another approach known as GGA+ U where U = 0.10 Ry is the Hubbard potential to affirm the electronic structure as GGA underestimates the band gap.The band profile from this approximation is pictured in same figure 3 which shows a significant change in band gap in spin down while metallicity is retained in spin up so, this approach also specified the half-metallic nature of the alloy.As GGA+ U is semi-empirical in nature [7] so not sufficient while probing the electronic structure Hence, we have also employed mBJ approximation.The band structures via this  approximation is also shown in same figure 3 which pictures the semiconducting nature of the alloy in both the spins as the energy levels are far from the fermi level in both channels with a band gap of (2.16 and 2.9) eV in up and down channels respectively.The value of band gap is more than 2 eV so, this alloy will be able to work at higher temperatures.Further, we have also analysed the densities of states (total and partial) which is in agreement with the band structure of the alloy.The total and partial density of states by mBJ is depicted in figures 4(a), (b) reflects the same nature of the alloy as by bands.The partial densities of states give information about the individual contribution of states which arrays that energy levels are far apart from fermi level hence, there is a gap in both the spins responsible for semiconducting nature of the alloy.To see the magnetism of SrPrO 3 alloy within the applied schemes of GGA, GGA+U and mBJ.The total magnetic moment and their individual presentation of the alloy are summarized in table 2. As reflected from the table 2 that magnetic moment of the alloy is 1.00 μB while the maximum contribution of magnetism comes from f-states of Pr.
Positive spin magnetic moments indicate the ferromagnetic interaction [23].

Mechanical properties
The elastic properties intrinsically define the mechanical properties of the material under the circumstances of external pressure executed on it.In this present study, we have calculated the three elastic constants i.e.C 11 , C 12 and C 44 to understand the mechanical stability [24] as the alloy has cubic nature, where C 11 is the longitudinal compression, C 12 explains transverse expansion and C 44 shear distortion.According to this model the three elastic constants for a cubic structure can be obtained and should fulfil the following relation defined below: For the case of SrPrO 3 alloy the stability outlook is reflected and enlisted in table 3. Thus, the non-negative values indicate that the alloy is elastically stable.In addition, the mechanical stability of these alloys has been fully testified by consuming the numerical values of elastic constants in several mathematical equations.Viogt-Reuss-Hill method is used to obtain the bulk modulus and shear modulus enumerated by the following equations; Bulk modulus (B) and Young's modulus (Y) explains the incompressible and stiffer nature of these compounds respectively.Poisson's ratio (ν) defines the plasticity of a compound.The value of Poisson's ratio for the alloy is 0.30.We have also calculated Zener's anisotropy factor (A) which indicates that the alloy is anisotropic in nature as its value is greater than 1 [25].To understand the nature of material, we have calculated the Pugh's ratio and Cauchy's pressure of which value is greater than 1.75.Hence, reveal the ductile nature of the material [26,27].

Thermodynamic properties
The study of thermal properties gives information about the impact of temperature and pressure on the alloy.
Here, we have calculated three different parameters namely, thermal expansion, specific heat capacity and Gruneisen parameter to investigate the thermal properties by using the quasi-harmonic Debye model within the temperature and pressure range of 0-800 K; 0-20 GPa respectively.Firstly, we have calculated the thermal expansion coefficient (α) portrayed in figure 5(a) which indicates that it increases slightly with increase in temperature while it decreases significantly with increase in pressure.It is due to the reason that as interatomic spacing decrease with the increase in pressure so there is a strong bonding between the constituents.The value of thermal expansion coefficient (α) at room temperature is 1.75 10 −5 K −1 .Next, we have calculated the specific heat C V ( ) as shown in figure 5(b) and from that we inferred that it increases with increase in temperature following T 3 law [28] and then achieve a constant value at higher temperature obeying the Dulong-Petit law [29].The value of C V at room temperature is 100.73Joule/mol/K.Further, we have determined the Gruneisen parameter (ϒ) which gives information about the anharmonicity in the crystal.The variation of Gruneisen parameter with temperature at different pressure points is shown in figure 5(c) which clearly indicates that the variation is almost changeless with change in temperature whereas with pressure it decreases notably.The value of ϒ at 0 GPa and 300 K is 1.96.

Thermoelectric properties
In the present age there is a surge in energy demand from the worldwide so it is important to find some different sources of energy other than conventional energy resources.In this category thermoelectric materials have drawn an extensive attention to work as an alternative to the regular source of energy.These materials have the ability to convert waste heat into electrical energy and this process is based on Seebeck effect, which provides the connection between voltage and the temperature gradient.The thermoelectric performance of a material is determined by figure of the band structure near the fermi level and the band gap is very useful in obtaining reliable transport properties in thermoelectric materials [30,31].Here, we have calculated the different transport parameters using the Boltzmann transport theory.These parameters namely Seebeck coefficient (S), electrical conductivity (σ/τ), thermal conductivity (κ), power factor and figure of merit (ZT) are calculated within a temperature range of 0-900 K.
At first, we have calculated the Seebeck coefficient, the variation of spin dependent Seebeck coefficient with temperature is depicted in figure 6(a) and from the graph we can see that the value of S for this alloy decreases with increase in temperature in both spin up and down channel suggesting the semiconducting nature of the alloy which is in accordance with the electronic properties.The difference between up and down graph for Seebeck coefficient lies on the fact that it is directly linked with density of states [32].We have also calculated the total Seebeck coefficient, which also follows the decreasing trend.It decreases because of the thermal excitations.The value of Seebeck coefficient at room temperature and at 900 K is given in table 4.
To measure the flow of electric current, we have calculated the electrical conductivity (σ/τ) of the alloy for both the spins.The variation with temperature is portrayed in figure 6(b) which reveals the semiconducting  For the alloy we have also determined the thermal conductivity which can be expressed as k where e is conductivity due to drift of electrons and holes while k is because of l phonon present in crystal.The graph of electronic and lattice part of thermal conductivity is pictured in figure 6(c) and from the graph we inferred that electrical thermal conductivity shows an increasing trend as charge carrier concentration increases with increase in temperature [33] while lattice thermal conductivity shows the decreasing trend due to surge in phonon scattering.
Additionally, we have computed the power factor, to test the applicability of material in thermoelectric domain.The variation is shown in figure 7(a) which shows that it increases with increase in temperature.This is due to the fact that it depends on Seebeck coefficient and electrical conductivity enumerated as: s t = PF S 2 as both these parameters increase with the temperatures it implies that the power factor also increases.
At last, we have calculated the figure of merit (ZT) directly from the power factor.The temperature dependence of figure of merit is sketched in figure 7(b) which shows an increasing trend for the alloy as it relies on power factor.The increasing value of power factor, electrical conductivity, ZT and decaying value of lattice thermal conductivity suggests that this alloy can have applications in solid state devices and thermoelectric realm.

Conclusions
Here, we have investigated SrPrO 3 alloy by using first principle calculations.Calculations reveal that this alloy is stable in ferromagnetic phase.Spin-resolved band structure confirms the semiconducting nature of the alloy.The mechanical figuration for the alloy shows ductile and anisotropic nature.The temperature and pressure variation of thermal expansion, specific heat and Gruneisen parameter within the temperature and pressure range (0-800 K; 0-20 GPa) has been discussed.Finally, evaluation of thermoelectric coefficients divulge that this alloy can be used in thermoelectric fields.

Figure 1 .
Figure 1.Structural arrangement of unit cell.

Figure 2 .
Figure 2. Optimization plots in FM and NM phases of SrPrO 3 alloy.

Figure 3 .
Figure 3. Band profile of SrPrO 3 alloy by GGA, GGA+U and mBJ, where arrow defines the possible spin states.

Figure 4 .
Figure 4. (a)-(b) Total (a) and partial densities (b) of states of SrPrO 3 alloy, where arrow defines the possible spin states.

Figure 5 .
Figure 5. (a)-(c) Variation of the thermal expansion coefficient, specific heat capacity and Gruneisen parameter with temperature of SrPrO 3 alloy.

Figure 6 .
Figure 6.(a)-(c) Variation of the Seebeck coefficient, electrical conductivity and thermal conductivity with temperature of SrPrO 3 alloy.

Table 4 .
The calculated value of the Seebeck coefficient (in microvolt/K) and electrical conductivity (in 10 20 Ω −1 m −1 s −1 ) for both the spins.thespins as it rises with increase in temperature.As from the graph we can see that for up and down spins the electrical conductivity variation is different this is because of the different band structures in up and down spins.It is due to the reason that with increase in temperature, more electrons can jump from the valence band to the conduction band resulting in increase in conductivity.The value of electrical conductivity for up and down spins at 300 K and 900 K is enlisted in table 4. We have also plotted total electrical conductivity depicted in same figure6(b)